FUNCTIONS OF TWO VARIABLES OF FINITE -VARIATION AND SUPERPOSITION OPERATORS
Abstract
Our paper is devoted to the description of the superposition operators which
map on function spaces of finite variation. We present the results which develop and generalize
the recent researches by J. Matkowski, J. Mis, D. Waterman and V.V. Chistyakov:
we introduce the notion of a total (two-dimensional) -variation for functions of two
real variables and show that the Waterman class of these functions with finite total variation
is a Banach space. Also, we give the description of the Lipschitzian superposition
(Nemytskii) operator mapping the Wateman class into itself.
map on function spaces of finite variation. We present the results which develop and generalize
the recent researches by J. Matkowski, J. Mis, D. Waterman and V.V. Chistyakov:
we introduce the notion of a total (two-dimensional) -variation for functions of two
real variables and show that the Waterman class of these functions with finite total variation
is a Banach space. Also, we give the description of the Lipschitzian superposition
(Nemytskii) operator mapping the Wateman class into itself.
About the Authors
Е. Громов
Национальный исследовательский университет - Высшая школа экономики в Нижнем Новгороде
Russian Federation
Russian Federation
О. Солычева
Национальный исследовательский университет - Высшая школа экономики в Нижнем Новгороде
Russian Federation
Russian Federation
В. Тютин
Национальный исследовательский университет - Высшая школа экономики в Нижнем Новгороде
Russian Federation
Russian Federation
References
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2. Chistyakov, V.V. Superposition operators in the algebra of functions of two variables with finite total variation // Monatsh. Math. 2002. V. 137. №2. P. 99-114.
3. Chistyakov, V.V., Solycheva O.M. Lipschitzian Operators of Substitution in the Algebra BV // J. of Diff. Equations and Appl. 2003. V. 9. №3/4. P. 407-416.
4. Dyachenko, M.I., Waterman D. Convergence of double Fourier series and W-classes // Trans. Amer. Math. Soc. 2004. V. 357. №1. P. 397-407.
5. Hildebrandt, T.H. Introduction to the theory of integration. New York and London: Academic press, 1963. 361 p.
6. Matkowski, J., Miś J. On a characterization of Lipschitzian operators of substitution in the BV[a,b] // Math. Nachr. 1984. V. 117. P. 155-159.
7. Waterman, D. On -bounded variation // Studia Math. 1976. V. 57. №1. P. 33-45.
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